Question

Piperazine is a diprotic weak base used as a corrosion inhibitor and an insecticide. Its ionization is described by the following equations. \(\mathrm{HN}\left(\mathrm{C}_{4} \mathrm{H}_{8}\right) \mathrm{NH}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons\) \(\left[\mathrm{HN}\left(\mathrm{C}_{4} \mathrm{H}_{8}\right) \mathrm{NH}_{2}\right]^{+}+\mathrm{OH}^{-} \quad \mathrm{p} K_{\mathrm{b}_{1}}=4.22\) \(\left[\mathrm{HN}\left(\mathrm{C}_{4} \mathrm{H}_{8}\right) \mathrm{NH}_{2}\right]^{+}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons\) \(\left[\mathrm{H}_{2} \mathrm{N}\left(\mathrm{C}_{4} \mathrm{H}_{8}\right) \mathrm{NH}_{2}\right]^{2+}+\mathrm{OH}^{-} \quad \mathrm{p} K_{\mathrm{b}_{2}}=8.67\) . The piperazine used commercially is a hexahydrate, \(\mathrm{C}_{4} \mathrm{H}_{10} \mathrm{N}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O} .\) A \(1.00-\mathrm{g}\) sample of this hexahydrate is dissolved in \(100.0 \mathrm{mL}\) of water and titrated with 0.500 M HCl. Sketch a titration curve for this titration, indicating (a) the initial \(\mathrm{pH} ;\) (b) the pH at the halfneutralization point of the first neutralization; (c) the volume of \(\mathrm{HCl}(\text { aq })\) required to reach the first equivalence point; (d) the pH at the first equivalence point; (e) the \(\mathrm{pH}\) at the point at which the second step of the neutralization is half-completed; (f) the volume of \(0.500 \mathrm{M} \mathrm{HCl}(\) aq) required to reach the second equivalence point of the titration; (g) the pH at the second equivalence point.

Short answer

Sketching the titration curve involves a nine-step process. The curve represents the titration of a diprotic base with a strong acid. The initial pH can be calculated using the first pKb and solving for [OH-] concentration. The first half-neutralization point occurs when [HB+] = [B], from which the pH can be calculated using the first pKb. The volume of HCl to reach the first equivalence point can be found using stoichiometry and knowledge about the initial concentration of the base. At the first equivalence point, the pH can be found using the second pKb and calculating the [OH-] concentration. The second half-neutralization point occurs when [A^2+] = [HB^+]. At this point, the pH can be calculated using the second pKb. The volume of HCl for the second equivalence point can be calculated using stoichiometry. The final pH (second equivalence point) can be computed considering the remaining H+ from the strong acid.

Step-by-step solution

Calculate the initial concentration

First, figure out the molar mass of the hexahydrate \( \mathrm{C}_{4} \mathrm{H}_{10} \mathrm{N}_{2} \cdot 6 \mathrm{H}_{2} \mathrm{O} \). Since 1.00g of the hexahydrate is dissolved in 100mL of water, you can calculate the initial concentration of the base, piperazine, in moles per liter.

Calculate the initial pH

For finding the initial pH, apply that the Kb of a base is given by \(Kb = [OH^-][HB^+]/[B] \). From this and given \(pKb_1 \) you will be able to estimate the [OH^−] concentration, and further the pOH and initial pH.

Calculate the pH at the first half-neutralization point

At the first half-neutralization point, [HB^+] = [B]. Solve for the pH using the expression for \(pKb_1 \).

Calculate the HCl volume for the first equivalence point

The equivalence point is reached when all of the base has been reacted with the acid. Use the initial concentration and the stoichiometry of the reaction to calculate the volume of 0.500 M HCl needed to reach this point.

Calculate the pH at the first equivalence point

As all the base is reacted, use \(pKb_2 \) and the expression \(Kb = [OH^-][A^2+]/[HB^+] \) to estimate the [OH−] concentration and further the pOH and pH at this point.

Calculate the pH at the second half-neutralization point

At the second half-neutralization point, [A^2+] = [HB^+]. Use \(pKb_2 \) and the equation for Kb to find the pH at this point.

Calculate the volume of HCl to reach the second equivalence point

As before, use stoichiometry of the reaction to calculate the volume of 0.500 M HCl to reach the second equivalence point.

Calculate the pH at the second equivalence point

For finding the pH at the second equivalence point, you need to remember that all the base is reacted at this point and estimate the H+ concentration remaining from the strong acid. Use this to calculate the pH.

Sketch the titration curve

For sketching the curve: On the x-axis plot the volume of HCl added, on the y-axis plot the pH. Mark the points from steps 2 to 8 on the curve.

Key concepts

Piperazine

Piperazine, a diprotic weak base, plays a significant role in both industrial and agricultural applications. It is commonly used as a corrosion inhibitor and insecticide. When considering its chemical behavior, it's essential to understand its ionization process.
Piperazine can accept protons in two stages due to its two nitrogen atoms acting as basic sites. This is represented by the given ionization equations:

• The first ionization: \( \mathrm{HN}(\mathrm{C}_{4} \mathrm{H}_{8})\mathrm{NH} \rightleftharpoons [\mathrm{HN}(\mathrm{C}_{4} \mathrm{H}_{8})\mathrm{NH}_{2}]^{+} + \mathrm{OH}^{-} \) with a \( pK_{b_{1}} = 4.22 \).
• The second ionization: \([\mathrm{HN}(\mathrm{C}_{4} \mathrm{H}_{8})\mathrm{NH}_{2}]^{+} \rightleftharpoons [\mathrm{H}_{2}\mathrm{N}(\mathrm{C}_{4} \mathrm{H}_{8})\mathrm{NH}_{2}]^{2+} + \mathrm{OH}^{-} \) with a \( pK_{b_{2}} = 8.67 \).

Understanding these stages is crucial for predicting the behavior of piperazine in acid-base titration scenarios, particularly during the titration curve sketching.

Equivalence Point

The equivalence point in a titration is critical because it marks where the amount of titrant added is stoichiometrically equivalent to the concentration of the substance being titrated.
In the case of the piperazine titration using HCl, there are two equivalence points due to its diprotic nature:

• The first equivalence point occurs when all of the \([\mathrm{HN}(\mathrm{C}_{4} \mathrm{H}_{8})\mathrm{NH}_{2}]^{+}\) has reacted to form \([\mathrm{H}_{2}\mathrm{N}(\mathrm{C}_{4} \mathrm{H}_{8})\mathrm{NH}_{2}]^{2+}\).
• The second equivalence point is reached when all piperazine molecules have donated both protons and all possible bases have been neutralized.

Reaching the equivalence points means that the pH of the solution shifts significantly. These points are shown on a titration curve, representing changes in pH against the titrant volume added. Understanding equivalence points is vital for correctly identifying the completion of reactions in titrations.

Neutralization Reaction

In an acid-base titration, a neutralization reaction occurs wherein an acid and a base react to form water and a salt.
This is the central process when titrating piperazine with hydrochloric acid (HCl). In this scenario:

• The base, piperazine, reacts with the strong acid, HCl.
• During the reaction, protons (\( \mathrm{H}^+ \)) from HCl interact with the hydroxide ions (\( \mathrm{OH}^- \)) generated by piperazine, producing water.

Such reactions are fundamental in detecting when equivalence points are reached and help determine the pH of the solution at various stages of the titration. Observing the neutralization process of piperazine provides crucial data for completing a titration curve accurately.

Weak Base

Piperazine is classified as a weak base, which means it does not fully ionize in aqueous solutions. This property affects its pH and makes its behavior distinct in titrations when compared to strong bases.
Since piperazine is a weak base, its degree of ionization is lower, influenced by its relatively higher \(pK_b\) values.

• The first dissociation constant: \( pK_{b_{1}} = 4.22 \), reflects moderate ionization in its initial stage.
• The second dissociation constant: \( pK_{b_{2}} = 8.67 \), is significantly reduced, indicating much lower ionization potential in the second stage.

With weak bases like piperazine, titration curves will often show gradual pH increases until the equivalence points, making it essential to accurately measure and calculate volumes and concentrations to determine precise pH levels.